IS : 4905 ·1968 (Reaffirmed 2001 ) Indian Standard METHODS FOR RANDOM SAMPLING ( Eighth Reprint DECEMBER 2002 ) UDC 519.271.3 :620.113 C Copyright 1969 BUI!EAU OF INDIAN STANDARDS MANAK BHAVAN. 9 BAHADUR SHAH ZAFAR MARG NEW DELHI 110002 Gr8 May 1969 IS: 4905· 1968 Indian Standard METHODS FOR RANDOM SAMPLING Methods of Sampling Sectional Committee, SMDC 4 Chairman DR A. V. SHRl SUKHATME Representing The Tata Iron & Steel Co Ltd, Jamshedpur Light Metals and Their Alloy Products Sectional Committee, SMDC 10, lSI Members V. D. AGARWAL SHRI O. P. MATHUR ( SHRI BHATTACHARJEE J. Alternate) Directorate General of Inspection (Ministry Defence) Indian Iron & Steel Co Ltd, Burnpur (Alternate) SOUTH Railway Board ( Ministry of Railways) of DR U. N. BHRANY SHRI A. C. ~1UKHERJEE CHEMIST & METALLURGIST, EASTERN RAILWAY, KHARAGPUR CHEMIST & METALLURGIST-I, RDSO, LUCKNOW ( Alternate) SHRI R. N. DATTA SHRI D. K. CHAKRAVARTY Directorate General of Ordnance Factories ( Ministry of Defence), Calcutta (Alternate) ( M uradnagar ) SHRI D. SEN ( Alternate) ( Katni ) SHRI S. B. FIRKE SHRI K. C. CHOUDHURI ( Copper and Copper Alloys Sectional Committee, SMDC II, lSI Alternate) ( (Lucknow) SHRI S. S. VAJDYANATHAN (Bombay) SHRI A. GUHA SHRI S. S. HONAVAR SHRt P. PATEL ( Alternate) Cast Iron and Malleable Cast Committee, SMDC 9, lSI Italab Private Ltd, Bombay Iron Sectional J. Alternate) Essen & Co, Bangalore; and Ores and Raw Materials Sectional Committee, SMDC 16, lSI Alternate) Foundry Sectional Committee, St\'IDC 17) lSI Indian Statistical Institute, Calcutta DR N.JAYARAMAN SHRI K. N. GURURAJACHAR ( SHRI R. M. KRISHNAN SHRI D. B. LAHIRI DR A. MATTHAI ( Alternate) SMRI Indian Non-ferrous Metals Manufacturers' Association, Calcutta SHRt M. 1\1. MOUDGILL (Alternat,) DR N. T. MATHEW Statistical Organization ( Ministry of Defence) SHRI S. P. AOARWALA (Alternate) SHRI A. K. MITRA Mitra S. K. Private Ltd, Calcutta SHRI M. N. MITRA ( Alternate) ( Continued on pag« 2 ) N. MAJUMDAR nUI{EAIJ {)F INDIAN STANDARDS MANAK BIIAVAN, 9 DAIIADUR SHAll ZAFAR MARG NEW DELIII 110002 IS : 4905 · 1968 ( Continuedfrom pag' 1 ) It/embers SHRt N. C. MITRA DR M. K. BOSE (Altlrnat') SHRI S. N. MUKER}I SHRI A. K. BHATTACHARJEE ( DR G. MUKHERJEE R,p"s,ntin, Government of India Mint ( Ministry of Finance) National Test House, Calcutta Alternate) Ferro Alloys Sectional Committee, SMDC 8, lSI; and Steel Tubes, Pipes and Fittings Sectional Committee, SMDC 22, lSI ' Refractories Sectional Committee, SMDC 18, lSI DR D. N. NANDI Steel Forgings Sectional Committee, SMDC 21, lSI SHRIA.PADMANABHAN Directorate General of Supplies & Disposals SHRI M. R. PATEL ( Inspection Wing ) ( Ministry of Works, Housing & Supply) SHRI B. B. BANERJEE ( Alternate) SHRI M. K. RAO Lead, Zinc, Tin, Antimony and Their Alloys Sectional Committee, SMDC 12, lSI SHRI D. K. RAY Wrought Steel Products Sectional Committee, SMDC 5, lSI SHRI G. V. D. UPADHYAYA Indian Bureau of Mines ( Ministry of Steel, Mines & Metals), Nagpur SHRI U. N. SARKAR ( Alternate ) SHRI S. VISWANATHAN Methods of Physical Tests Sectional Committee, SMDC 3, lSI DR B. N. SINOH, Director Genera], lSI (Ex-o.fficio klemher) Director ( Stat) Secretary SHRI Y. K. BRAT Deputy Director ( Stat), lSI Panel for Random Sarnpling, Convener S~fDC 4 : P7 DR D. SINGH Institute of Agricultural Research Statistics ( leAR ), New Ddhi Statistical Organization ( Ministry of Defence) Indian Statistical Institute, Calcutta Members DR N. T. l\lATHEw SHRI A. S. Roy 2 AMENDMENT NO.1 JULY 1992 TO IS 4905: 1968 METHODS FOR RANDOM SAMPLING (Page 14, Example 14, line 6) - Substitute 'less than or equal to 999' for 'less than 999'. (page 14, Example 14, last sentence) existing sentence: Substitute the following for the 'This procedure will result in the rejection of very few numbers- those above 489 and less than or equal to 500 and again tbose above 989 and less than or equal to 999 (which give remainders greater than 489 when divided by 500 ), thereby leading to the rejection of only 21 numbers as compared to 510 by the earlier procedure.' (MSD3 ) Printed at Simco Printing Press. Delhi IS : 4905· 1968 Indian Standard METHODS FOR RANDOM SAMPLING o. FOR E WORD 0.1 This Indian Standard was adopted by the Indian Standards Institution on 20 December 1968, after the draft finalized by the Methods of Sampling Sectional Committee had been approved by the Structural and Metals Division Council. 0.2 Sampling is of fundamental importance for estimating the quality of a lot or ascertaining its conformity to the requirements of a specification. The economy, reliability and practicability of the sampling procedures have made them almost indispensable in most of the industrial and trade applications. However, the reliability of the conclusions drawn on the basis of the sample depends on its representativeness and the method of its selection. Since this vital aspect of sampling has not received wide attention, it was felt desirable to lay down the basic procedures for the selection of a random sample under diverse situations. It is hoped that the sampling methods as laid down in this standard, when implemented, would ensure a truly random and representative sample leading to sound and satisfactory estimation of lot quality. 0.3 This standard is one of a series of Indian Standards relating to techniques of statistical quality control. Other standards published so far in the series are: IS : 397-1952 Method for statistical quality control during production by the use of con trol chart IS: 1548-1960 Manual on basic principles of lot sampling IS : 2500 ( Part I )-1963 Sampling inspection tables: Part I Inspection by attributes and by count of defects IS: 2500 ( Part II )-1965 Sampling inspection tables: Part II Inspection by variables for percent defective IS : 5002 .. 1969 Method for determination of the sample size to estimate the average quality of a lot or a process 0.4 In preparing this standard, considerable assistance has been derived from the following publications: COCHRAN (W G). Sampling techniques. John Wiley & Sons, Inc, New York. DEMING ( WE). Some theory of sampling. John Wiley & Sons, Inc, New York. 3 IS : 4905 · 1968 M H ), HURWITZ ( W N ) and MADOW (W G). Sampling survey methods and theory. Vol I Methods and applications, Vol II Theory. John Wiley & Sons, Inc, New York. SUKHATME (P V). Sampling theory of surveys with applications. Indian Society of Agricultural Statistics, New Delhi. A million random digits with 100 000 normal deviates. 1955. Rand Corporation, Illinois, USA. 0.5 In reporting the result of a test or analysis, if the final value, observed or calculated, is to be rounded off, it shall be done in accordance with IS : 2-1960*. HANSEN ( 1. SCOPE 1.1 This standard lays down general procedures for the selection of items from a lot on an objective basis by using random sampling techniques. It also describes the methods of calculation of simple estimates like mean and proportion of defective from the sample data. 2. TERMINOLOGY 2.0 For the purpose of this standard, the following definitions shall apply. 2.1 Item - Ultimate unit of product or material which is to be selected. 2.2 Lot (Population) - Totality of items or individuals under consideration. 2.3 Lot Size ( N) ~ Total number of items in the lot. 2.4 Sample - Collection of items selected from the lot ( or the population). 2.5 Sample Size ( 11, ) - Number of items in the sample. 2.6 Sampling Fraction (~ ) - Ratio of sample size to the lot size. 2.7 Random Sampling - A procedure of selection in which the chance for the inclusion of any item in the sample is predetermined. 2.8 Defective - An item the quality of which does not meet the specified requirement. 3. RANDOM SAMPLING METHODS 3.1lSimple Random Sampling 3.1.0 In case the lot consists of a number of items such that each item is easily identifiable and, apart from the lot size, no other information about the composition of the lot is available, the method of simple random sampling may be followed for selecting the items for the sample. ·Rules for rounding off numerical values ( r,vised ). 4 IS : 4905 · 1968 3.1.1 According to this method, the sample of the requisite size n is drawn from a lot of size Nin such a manner that, while selecting an item, the chance for any item of the lot being included in the sample is the same. An item once drawn is not placed back in the lot. NOTE - In case the item drawn is put back in the lot before the next item is selected, thereby allowing for the possibility of the same item being chosen more than once for inclusion in the sample, the method is usually referred to as simple random sampling with replacement. However, this method of sampling is not commonly used in industrial practice and hence it has been left out from further consideration in this standard. 3.1.2 For the selection of a simple random sample of n items from a lot of N, the first requisite is to obtain n random numbers ( see 4 ) which lie in the range 1 to N. For this purpose, starting from any number of the random number table (see Appendix B) and continuing on with the numerals in any direction, right or left, up or down, the succeeding numerals are copied out one-by-one till n different numerals are obtained. The numerals zero or those which are greater than N or which have already occurred, shall be omitted. The numerals noted down in this manner shall then be arranged in the ascending order of magnitude. Starting from any item in the lot and counting them in one order, the items corresponding to the numerals already noted down shall be withdrawn to constitute the required sample of size n. Example 1: It is desired to obtain a sample of 10 electrical components from a lot of 200. If the components in the lot are mentally-assigned serial numbers up to 200, the problem then becomes to obtain 10 random numbers in the range 1 to 200. Taking the 3-digited random numbers from Appendix B and starting from any number, say 149, occurring in the eleventh row and 11, 12 and 13 columns on page 19 and proceeding downwards, the numerals less than 200 are noted down. Thus the 10 numerals so obtained are 149, 62, 174, 177, 142, Ill, 165,13,17 and 194. When arranged in ascending order of magnitude, the numerals become 13, 17, 62, Ill, 142, 149, 165, 174, 177 and 194. The electrical components in the lot corresponding to these numbers shall then be selected to constitute the required random sample of size 10. 3.1.2.1 The procedure as given in 3.1.2 may however result in the rejection of a large proportion of the random numbers which exceed the lot size. Hence, an alternative and more convenient method for the selection of random numbers is given in 4.3. 3.1.3 For a simple random sample of size n, the estimate of the average quality of a lot may be obtained by dividing the sum of the sample test results or observations by the sample size (see also A-I.I ). 5 I8t_5.1968 Example 2: For the 10 electrical components selected in Example 1, if the resistance is obtained as 639, 640, 650, 647, 662, 637, 652, 643, 657 and 6490, then ~ ) 639 + + 649 rv 6 476 647 6 Average ( ~ = . ~" = - - = ·Q 10 10' Hence 647'6Q is the estimate of the mean resistance for all the items in the lot. · 3.1.4 The estimate of proportion of defectives in the lot may be obtained by dividing the total number of defectives as observed in the sample by the size of the sample ( see also#A-l.2 ). Example 3: If the maximum limit for resistance of the component in Example 2 is specified as 655Q, then there will be two defectives out of 10 for the selected sample. Hence, the estimate of proportion of defectives in the lot is given by iJ.o :; 0'2 or 20 percent. 3.2 Stratified SampUng 3.2.0 When a lot c.onsists of items which can be divided into a certain number of more homogeneous groups or strata, the method of stratified random sampling may be followed according to which each group of stratum is sampled separately so as to obtain a sample representative of the entire lot. In such cases, this method of sampling may be generally more efficient than the random sampling as the latter may not always result in the selection of the items from each stratum of the lot, thereby affecting the representativeness of the sample drawn. 3.2.1 The application of the stratified sampling method would require the division of a lot into a suitable number of strata and then the selection of a simple random sample from each of the stratum to make up the desired sample size. For this purpose, the division of a lot into the strata may be undertaken on the basis of.fhe homogeneity of the items wi.thin a lot, convenience of sampling or such other considerations which would make the items within each stratum 'as much alike as possible whereas those between the strata may be as much different as possible. The allocation of the number of items to be selected from each stratum is sometimes done on the basis of the variability of the items within a stratum. But in most of the industrial applications such a knowledge is hardly available in advance and hence the number of items to be selected from the stratum is generally taken to be proportionate to the stratum size, that is, the number of items in the stratum. This procedure known as proportional allocation would have the added advantage of considerably simplifying the estimation of the lot mean or proportion of defectives. I t would also be advisable to ensure that a minimum of two items are selected from each stratum. 6 IS : 4905 - 1968 3.2.2 The selection of the sample items from each of the stratum shall be done on the same lines as given in 3.1.2. Example 4: Suppose a lot of 300 spanners manufactured in three different shifts are packed in three separate boxes of 125, 100 and 75. If a sample of 12 spanners is to be selected from the lot then the three different boxes may be treated as three different strata and 5, 4 and 3 spanners ( as obtained by distributing 12 in proportion to the sizes of the 3 boxes) may be selected from the corresponding three boxes. The actual selection of sample items frorn each stratum would be made as described in Example 1. 3.2.3 For a stratified random sample of size n, the estimate of the lot mean quality will be calculated by dividing the sum of observations from all the strata by the sample size ( see also A-2.1 ). Example 5: If for the 12 spanners selected in Example 4, the hardness ( on the Rockwell hardness scale) is obtained as: 39, 39, 42, 45, 43 for the items from the first stratum; 37, 40, 41, 38 for the items from the second stratum; and 40, 38, 40 for the items from the third stratum; the sum of the observations for the three strata are obtained as 208, 156 and 118. Hence, the estimate of the mean hardness for the lot 208 + 156 + I 18 = 12 482 :..... 40-17 12 · 3.2.4 The estimate of proportion of defectives in the lot may be calculated by dividing the total number of defectives found in all the strata by the sample size ( see also A-2.2 ). Example 6: If in Example 5, the specified hardness range for the spanners is given as 38 to 42, then the number of spanners not conforming to the hardness requirements are 2, 1 and 0 respectively for the items selected from the three strata. Hence the estimate of the proportion of spanners not conforming to the hardness requirement in the lot = 2+1+0 12 3 = 12 = 0·25 or 25 percent. 7 IS : 4905 · 1968 3.3 Systematic SampUag 3.3.0 When the items in a lot are presented in an orderly manner ( such as piles of asbestos sheets or stacks of cement bags) it is possible to considerably simplify the selection of a random sample of the required size. Instead of choosing the desired number of random numerals and then drawing the items corresponding to these numerals as illustrated in simple random sampling ( see 3.1.2 ), one item is chosen at random from the lot and thereafter the items are selected regularly at predetermined intervals. I t has been established that this method of systematic sampling is quite a good approximation to the simple random sampling method described earlier, provided there is no deliberate attempt to manipulate the sequence of the items in the lot in any desired manner while the lot is presented for inspection. Because of its simplicity of operation and the appealing nature of the ' spread' of the sample items all through the lot, the systematic sampling has found a very wide application in industry as well as in other fields like agricultural and socio-economic surveys. 3.3.1 The method consists of first selecting a single sample item from the population of N items and thereafter selecting items at regular predetermined intervals to make up the desired sample of size n, For this purpose, the integral part of NJn ( say r) is taken as the interval and then the items are counted in one order and every rth item thus counted is withdrawn until the sample of required size is obtained. Example 7: Suppose a lot of 250 containers of baby food are stored neatly in racks and a sample of 8 containers is to be selected at random. Calculate the integral part of It.Q ( 31·25 ) which is 31. Starting from any container (for the choosing of which the help of random number tables may be sought, if necessary), count the containers in one order as 1, 2, 3, , 31 and so on. Every 31st container so counted shall be chosen till a total of 8 containers are obtained to constitute the desired sample. In this c-ase, the sample containers to be selected are those corresponding to the serial numbers 31, 62, 93, 124, 155, 186,217 and 248. 3.3.2 For a systematic sample of size n the estimate of the lot average or the proportion of defective in the lot may be worked out using the relevant formulae given in A-3.1 and A-3.2 which are similar to those for simple random sampling (see 3.1.3 and 3.1.4). 3.4 Cluster SampUng 3.4.0 When the lot submitted for inspection consists of certain groups of clusters of items, it is sometimes advantageous and economical to select a few 8 IS: 4905· 1968 clusters and then examine all the items in the selected clusters. This would be the case, for example, when the lot consists of items packed in cartons and it is either impracticable or costly to repack the cartons opened for selecting sample items, The method has also extensive applications in agricultural andsocio-economic surveys, In the former case, for example, if the problem is to estimate the area under a certain crop in a district, it may be simpler and more economical to ascertain the area under crop in all fields of a selected village ( cluster) rather than a few fields in each of the villages of the district. It may however be mentioned here that unlike the stratified sampling, satisfactory results for cluster sampling would be obtained when the items within a cluster are quite heterogeneous, 3.4.1 The method consists of selecting a few of the clusters at random without replacement in the first instance. Thereafter, all the items in each of the selected clusters are pooled to obtain the required sample from the lot. 3.4.2 The selection of the sample clusters from the lot shall be done on the same lines as given in 3.1.2_ Example 8: In a firm producing BHC technical on a batch process, normally 3 to 5 batches are manufactured in a day, Considering all the batches manufactured in a day as a cluster, if it is decided to sample 5 clusters from a total of 50 obtained during a period of 2 months, then the procedure for the selection of these 5 clusters shall be the same as indicated in Example 1. All the batches produced on each of the selected days ( clusters) shall then be tested. 3.4.3 For a random cluster sampling the estimate of the lot mean may be calculated by using the formula given in A-4.1_ Example 9: Let the 5 clusters chosen in Example 8 consist of 5, 4, 5, 2 and 3 batches respectively, Let the hydrolysable chlorine content (in percentage) for the different batches of the selected clusters be obtained as: Total 35-8, 35·6, 34-6, 35-9, 36-0 35-9, 35-9, 36-2, 35-2 36-0, 36-3, 36-1, 35-9, 35-8 36-1, 36-3 177-9 143-2 180-1 34-8, 35-5, 35-6 9 72-4105-9 IS: 4905 -1968 If it is known that the total number of 50 clusters consisted of 188 batches in all, then the estimate of the average hydrolysable chlorine con ten t for the en tire process for the period under reference is obtained as ~~!! x ( 177·9 + 143·2 + 180·1 188 + 72·4 -t- 105·9) 50 679·5 188 - = 36·1 percent. =-x 5 3.4.3.1 The above method of estimating the lot mean has one drawback in the sense that it needs the knowledge of the total number of items in all the clusters of the lot. Quite often, it so happens that the number of items in the selected cluster alone is available because of the complete enumeration of the items therein. In such situations the lot mean can be estimated fairly accurately by dividing the sum of the observations of all the items in the selected clusters by the total number of items in the selected clusters. Thus in the Example 9, if the information regarding total number of batches in all the 50 clusters in the lot is ignored, then the estimate of the lot mean is obtained as 177·9 + 143·2 + 180-1 + 72-4 + 105·9 679'5 5 + 4 + 5 + 2 + 3 ..~-- = 19- = 35'8 percent. NOTE - If all the clusters are of the same size, the estimate of lot mean under both the situations described above is obtained by dividing the stun of all the observations in the selected clusters by the total number of items in the selected dusters (which is nothing but the product of the cluster size and number of clusters selected ). 3.4.4 The estimate of the proportion of defectives in the lot may be calculated by using the formula given in A-4.2. Example 10: If in the Example 9 the minimum hydrolysable chlorine content for BHC technical is specified as 35·5 percent, then the number of defective batches in the 5 selected clusters of Example 9 are obtained as 1, 1, 0, 0 and 1 respectively. Hence, the estimate of the proportion of defectives in the entire production during the period under reference is given by I-l)Jl ( 1 + 1+ 0+ 0 + 188 1) = '-WS- fl.,,!! X 3 = 1878- 0-16 or 3 16 percent. 3.4.4.1 If the information regarding the total number of items in all the clusters in the lot is not readily available, then the estimate of the proportion of defectives can be obtained by dividing the number of the defective items in all the selected clusters by the total number of items in the selected clusters, 10 IS: 4905 -1968 Thus in Example 10, the estimate of the proportion of defectives in the lot as obtained by ignoring the total number of batches in the lot, is 1+1+0+0+.1 _ 3 5 +4+ 5'~-+ 2 + 3 - -19-== 0·16 or 16 percent. NOTE - If all the clusters are of the same size, the estimate of the proportion of defectives in the lot under both the situations described above is obtained by dividing the number of d- fectivc items in the selected clusters hv the total number of items in the selected clusters. ' 3.5 Two-Stage Sampling 3.5.0 When a lot submitted for inspection consists of a large number of packages each consisting ofa number of items, it may not be quite economical and feasible to open each of the packages for drawing sample items ( as in the case of stratified sampling described in 3.2 ), or to open only a few packages and inspect all the items in these packages ( as in the case of cluster sampling described in 3.4). In such cases, it may be desirable to first select an adequate number of packages and then to choose the necessary number of items from each of these selected packages. Because of the two stages involved in this method of sampling it is referred to as two-stage sampling. The first and second stage units are also sometimes called as "priruary ' and , ultimate' units. 3.5.1 The method consists in selecting the items for the sample in two stages; in the first stage a desired number of primary units is selected at random and in the second stage, the required number of items are chosen at random from the selected primary units. 3.5.2 The selection of the primary units in the first stage as also the selection of items from the chosen primary units is done on the same lines as described in 3.1.2. Example 11: Suppose a certain variety of cotton saris, each of 5 metre length is packed in bales and a lot consisting of 8 bales is submitted for inspection. Let the number of pieces in the 8 bales be 45, 50, 55, 43~ 58, 60, 48, and 41 respectively (a total of 400 pieces in all ). If it is intended to estimate the warpway breaking load of the pieces in the lot, then a two-stage random sampling method may be employed, In the first stage, if 3 out of the total of 8 bales (say second, sixth and eighth ) are selected, then from the selected bales 3, 4 and 2 pieces Inay be chosen respectively so as to obtain a sample of size 9. 3.5.3 For the two-stage sampling, the estimate of the lot mean may be obtained by using the formula given in A-S.t. 11 IS I 4905 · 1968 Example 12: If the breaking load (in kilograms) of the 9 selected pieces in Example 11 is obtained as 52, 51, 49 for sample pieces from the second bale 48, 53, 53, 50 for sample pieces from the sixth bale 54, 55 for sample pieces from the eighth bale then the estimate of the mean value of the breaking load for the 3 bales are obtained as 50- 7, 51-0 and 54·5 kg respectively. The estimate of the mean breaking strength of all the pieces in the lot is then obtained as t (50 X 50'7 + 60 400 X 51'0 + 41 X 54'5) = !-~8292= 52'2 kg. . 3 x 400 3.5.3.1 The above method of estimating the lot mean has one drawback in the sense that it needs the knowledge of the total number of items in the lot. However, it so happens frequently that the information regarding the number of items is available only for the selected primary units but not for the others unless some extra effort is put in, Insuch cases the lot mean may be estimated fairly accurately by suitably weighting the means of the selected primary uni ts by their respective sizes, Thus in the Example 12, if the information regarding number of pieces in the lot is not available, then the lot mean is estimated as 50 X 50-7 + 60 50 + 60 + 41 X 51·0. + 41 X 54·5 == 7829-5 151 == 51-9 k g. NOTE - If the sampling fraction remains the same for all the selected primary units ( that is, the number of items selected are proportional to the size of the primary units), then the estimate of the lot mean in this case is simply obtained r>y dividing the sum of all the sample observations by the total number of sample items selected, 3.5.4 The estimate of the proportion of defectives in the lot may be calculated by using the formula given in A-S.2. Example 13: If the minimum breaking load for the fabric is specified as 50 kg in Example 12, then the estimate of the proportion of defective pieces in the three selected bales are obtained as i, i and 0 respectively. Hence the estimate of the proportion of defectives in the lot is given by f (50 X ! + 60 X i + 41 X 12 0) 400 400 = 0-21 or 21 percent. X == '3 8 31·7 IS: 4905· 1968 3.5.4.1 If the information regarding the total number of items in the lot is not readily available, then the estimate of the proportion of defectives may be obtained by suitably weighting the proportion of defectives in the selected primary units by their respective sizes. Thus in Example 13, the estimate of the proportion of defectives in the lot as obtained by ignoring the total number of items in the lot is ---5-0-+-6-0-+-4-1-----~-~ 50 X 1 + 60 X 1+ 41 X 0 = 31·7 151 =0·21 or 21 percent. NOTE - If the sampling fraction remains the same for all the selected primary units, then the estimate of the proportion of defectives in this case is simply obtained by dividing the number of defectives found by the total number of items sampled. 4. RANDOM NUMBERS 4.1 Appendix B contains 30 000 digits ( from 0 to 9 ), arranged randomly on 15 pages. For convenience of reading, the digits are grouped into sets of 5 and are arranged in 40 rows and 50 columns on each page. 4.2 Selection of Random Numbers - For obtaining random numbers, anyone of the 15 pages may be used. After choosing a page of random numbers, a pencil may be dropped blindly and the starting point for reading off the numbers shall be the random digit( s) nearest to the spot indicated by the pencil. The random number may then be read up or down, right or left but it may perhaps be more convenient to read them vertically downward till the bottom of the page is reached and thereafter the numbers in the adjacent column( s) to the right are read starting from the top of the page and moving downward as earlier. This systematic method of reading the numbers may be carried on to the succeeding pages if necessary. 4.2.1 If the random numbers to be chosen consist of single digit ( as would be the case when lot size does not exceed 10) then the digit of any single column may be used, taking 0 to represent 10. 4.2.2 If the random numbers to be chosen consist of two digits ( as would be the case when lot size does not exceed 100) then any two adjoining columns may be used, taking 00 to represent 100. 4.2.3 If the random numbers to he chosen consist of three digits (as would be the case when the lot size does not exceed 1 000 ) then any three adjoining c.olumns may be used, taking 000 to represent 1 000. 4.2.4 If the random numbers to be chosen consist of four or more digits then a similar method as outlined above may be followed. 4.3 Simplifying Techniques in the Selection of Random Numbers When a large number of random numbers are to be chosen and the total 13 IS : 4905 - 1968 number of i terns in the lot is not an integral power of 10 ( such as 100, 1 000, 10 000, etc) then a simplifying technique, which is illustrated in Example 14 may be helpful. Example 14: Suppose the lot consists of 489 items and it is intended to select 75 items at random, one method would be to select the three-digited random numbers from 001 to 489 and choose those items corresponding to the selected numbers, But this procedure may be rather wasteful as it will result in the rejection of all three-digited random numbers which are more than 489 and less than 999, that is 510 in all. Hence, more than 50 percent of the selected numbers will have to be discarded. In order to reduce the rejection of random numbers, an alternative method would be to choose the three-digited random numbers, divide them by a convenient round figure larger than the lot size, say 500, and then choose the items corresponding to the remainder obtained. This procedure will result in the rejection of very few numbers - those above 489 and below 500 and again those above 989 and below 999 (which give remainders greater than 489 when divided by 500), thereby leading to the rejection of only 22 numbers as compared to 510 by the earlier procedure. APPENDIX A ( ~lauses 3.1.3,3.1.4,3.2.3,3.2.4,3.3.2,3.4.3, 3.4.4, 3.5.3 and 3.5.4 ) FORMULAE FOR THE ESTIMATION OF LOT AVERAGE AND LOT PROPORTION OF DEFECTIVES A-O. For the different sampling methods covered in the standard formulae for the estimation of lot average (when the sample data is of the quantitative type) and the lot proportion of defectives (when the sample data is of the qualitative type) are given in A-I to A-5. A-I. SIMPLE RANDOM SAMPLING A-I.I If Xl' x 2 , ·············· ·XfI are the test results or observations corresponding to the n sample items selected from the lot of size N by the simple random sampling without replacement then the sample average (f) is calculated as n x L XI i= 1 n = Xl + x:a+ " n ! is also an estimate of the average ( or mean) quality for the lot. 14 IS : 4905 · 1968 A-l.2 If out of the n sample items selected from a lot of size N, a total of d defectives are observed, then proportion of defectives (p) in the sample is calculated as -~-. n d p is also an estimate of the proportion of defectives in the lot. A-2. STRATIFIED SAMPLING A-2.1 Suppose a lot of size N consist of k strata comprising of Nt, Nt, . Nt items so that N J XI + + N k = N. + II. II ··· Let n., n2, nk items be selected from k strata by the stratified random sampling method so as to obtain sample of size n ( = n1 n. + nk)· If the test results or observations corresponding to the sample of size n are ......... + + XII' XiS' ···· · ···· '.'X I X 21, X 22' ············ X 2 "1 R2 for the n1 items from the first stratum, for the n2 items from the second stratum, ............................................................................................ Xkh ""kl'.·'· Xkra k for the nk items from the kth stratum then the average for the various strata are calculated as!h x!, !~ The estimate of the lot average quality is then obtained as N] Xl + N S ! 2 + Nk"f k + N If the sampling fraction is same for all the strata, that is = - = t say Nk ' then the estimate of the lot average quality is simply obtained as XII t - J =_ NI N2 n n nk + +Xkft ----t--:N~--- k _ XII + n +Xk" - k A-2.2 If the number of defectives found in the samples from the k strata are d1, d., dk then the estimates of proportion of defectives in the various strata are calculated as Pl (= ~: ), P. (= ::) , + .IV15 It (== ~: ) The estimate of the proportion of defectives in the lot is calculated as NIP! + NsPs + N1c pt IS : 4905 - 1968 If the sampling fraction is same (say t) for all the strata, then the estimate of proportion of defectives in the lot is obtained as d) -t- d2 + dk _ d, + d2 + dk tN n A-3. SYSTEMATIC SAMPLING A-3.1 If Xl' X2~ ········ 'X n are the test results or observations corresponding to the n items selected from a lot of size JV by the systematic sampling method, then the sample average (x) is calculated as Zf = Xl + X 2 + Xn + n i is also an estimate of the average quality characteristic for the lot. A-3.2 If out of n sample items selected from a lot of size N, a total of d defectives are observed, then the proportion of defectives (p ) in the sample is d calculated as -". n p is also an estimate of the proportion of defectives in the lot. A-4. CLUSTER SAMPLING A-4.1 Let a lot consist of k clusters of sizes N I , N 2 , NI + N2 + ··········· · Nk = N. N k so that Suppose r clusters out of the k clusters in the lot are selected so that the total number of items in the r selected clusters is 11. If the test results or observations corresponding to the jth selected cluster of size N J are given by XiI' Xi2' ······ Xj}/., the sample mean for thejth cluster is calculated as J », = Xi] + Xj2 + N i + XJX i The estimate of the lot mean quality is then obtained as r k ~ X J xJ --X--. r A-4.1.1 If the information about the total number of clusters in the lot ( k) or the lot size (N) is not available, then the estimate of the lot mean quality may be obtained as , r I: N; xJ- =--_._~ Xi XJ r n :t N; 16 IS : 4905 · 1968 A-4.2 If the Jth selected cluster has d, defectives, then the proportion of defectives in the jth cluster is given by PJ ( = ;J ) The estimate of the proportion of defectives in the lot is then calculated as r r k L NiP} k r N = r -- .N ~ d· A-4.2.1 If the information about the total number of clusters in the lot ( k) or the lot size ( N) is not available, then the estimate of the proportion of defectives in the lot may be obtained as r ~ r NiP; r _ ~ dJ 11 L XJ NOTE - If all the clusters are of the same size, we have N1 = X z = .. r = Nk c::: N T Then the estimate of the lot average quality is simply obtained as ~;i r Also the estimate of the proportion of defectives in the lot reduces to r 1., Pi I dJ -=a-r n A-5. TWO-STAGE SAMPLING A-S.t Let a lot consist of k primary units which in turn comprise of N h N I J ................. .N k items so that N I + N 2 + + N k = N. Suppose r primary units are selected from the lot at random and from each of these r primary units suitable number of items are chosen so that the total number of items obtained is n. If from the jth primary unit selected, nJ items are chosen and the test results or observations corresponding to l ············xJn these n; items are given by XII' Xii' then the sample average for thejth primary unit is given by XJ = XII + XJI + "i + XJ,. I 17 IS c 4905 · 1968 The estimate of the lot average quality is then calculated as , ·r -----xA-S.I.! If the information on the total number of primary units in the lot (k ) or the total number of items in the lot ( .N) is not available, then the estimate of the lot mean may be obtained as r ~ k ~ N/l j X/Xi r ~ s, A-5.2 If the "1 second-stage units selected from the jth primary unit contain dl defectives, the proportion of defectives in thejth primary unit is obtained as Pi [= :; ] k I: Xi Pi The estimate of the proportion of defectives in the lot is then taken as , r - N- A-S.2.1 If the information regarding the total number of primary units ( k) in the lot or the lot size ( .N) is not available, then the estimate of the proportion of defectives in the lot may be obtained as r :I; N; Pi r ~ XI NOTa - If the sampling fraction is the same for all the r primary units selected, then the estimate of the lot mean quality as obtained from A-S.I.I is r r ~;n;i; , XIII == In;xl n Similarly, the estimate of the proportion of defective u obtained from '\-5.2.1 is , , , r I n;/JI == 1: nJPS .. Xnl n I dt n 18 IS : 4905 · 1968 APPENDIX ( Clause 4.1 ) B TABLE OF RANDOM NUMBERS Col No. Row No. I 2 3 1-5 6-10 11-15 16-20 21-25 26·30 31-35 36-40 41..45 46-50 45 6 7 8 9 ]0 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 '27 28 29 30 31 32 33 34 35 36 37 38 39 40 32 533 76520 13 586 34 673 04805 64 894 74296 24805 68 953 19645 09 303 23 209 02529 09376 70715 38311 99970 80 lSi 36 147 64 032 74 717 34072 76850 36697 10805 4557182406 35303 85269 77f02 0205165692 68665 63 573 32] 35 05 325 47 048 90 553 73 796 45 753 03 529 64 77R 35 808 98 520 17 767 14 905 68 607 22 109 11 805 05431 39808 27 732 50 725 8345299634 06288 98083 13746 88685 40200 86507 58401 36 766 99 594 67 348 87 517 64969 91 826 65 481 17 674 17468 50950 58047 80 124 35 635 17 727 08015 45 318 74350 99817 77402 77214 43236 69916 26803 66252 29 148 36936 09893 20505 14- 225 68 514 46127 91 499 14523 68479 27686 46 162 80 336 94 598 26 940 36 858 70 297 44104 81949 8~157 47954 32979 12 550 73 742 11 100 02040 12 860 63 G06 49329 16505 34404 40219 61 196 90446 26457 47 774 51 924 15474 45 266 95 270 79953 59 367 94557 28 573 67 897 54 387 54 622 42481 16213 97344 08721 16868 23523 78317 7320B 89837 68 935 04493 52 494 75246 33824 45862 00549 97 654 64 051 88 159 96 119 35 963 15 307 26898 09354 33 351 59808 08 391 45427 26842 83609 46058 85 236 01 390 92 286 77 281 32 179 00 597 87 379 25 241 05 567 69234 61 406 20 117 45 204 15956 19 565 41 430 01 758 75 379 40419 45 155 14938 19476 07 246 43667 94864 31 994 36 168 10851 34888 10 097 37 542 08 422 99019 12807 66065 31060 54 876 80 959 09 117 39 292 74 945 24037 20636 10402 00822 91 665 02 560 15953 34 764 35 080 33 606 31165 88676 74397 04436 27659 36653 98951 16877 12 171 76833 36 170 65813 39885 11 199 29 170 4261486799 07439 23403 09732 74818 73053852471862388579 57 548 28 468 28 709 83491 25 624 34 282 60 935 20 344 35 273 88 435 40 558 60 970 93 433 50 500 73 998 6824·8 29405 24201 52 775 67851 70078 18475 40610 68711 77817 67 951 90364- 76493 29609 11 062 08928 93 785 61 368 23478 34 113 76974 73039 57 186 40218 1654422 374 21 115 78 253 14385 53 763 00210 45521 64237 96286 02655 87 203 76621 13990 94400 56418 56 788 96297 78822 54382 14598 83554 94750 89923 370B9 20048 34 135 53 140 33 340 42 050 82 341 26575 57600 4088122222 06413 74 697 96644 89439 28 707 25815 52563 43651 77082 07207 31 790 33 729 6:> 394- 59593 42 582 60 527 83848 82 396 10 118 33 211 59466 44431 91 190 42 592 92 927 45973 4876703071 12059 2570146670 91 416 26252 29663 05522 82 562 51 025 61 962 79 335 65 337 12 472 63896 54692 82 391 23287 29529 35462 77974 50024 90 103 39333 49 700 13021 24 892 78565 20 106 44077 93910 83647 70617 42 941 07 007 86 743 17 157 85 394 II 838 60000 18 743 92423 97 118 96338 21 585 66 674 36806 84962 85207 94543 59047 90033 20826 69541 81 553 01 540 35456 05014 51 176 ( Continrud ) 19 IS : 4905 · 1968 (~ol No. Ro'w 1-5 6-10 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46-50 No. 4) 42 98086 33 185 80951 79752 18633 74 029 54 178 11 664 48 324 69074 09 188 90045 73 189 75 768 54016 08 358 28 306 53 840 91 757 89 415 77 513 19502 21818 51 474 99559 33 713 85 274 B4 133 56 732 65 138 38001 37402 97 125 21826 73 135 24826 16232 00406 49 140 32537 43 902 45611 49883 77928 94 138 20097 85497 50207 76490 44056 69910 03 264 86233 53 741 9~ 694 03820 37174 59313 66499 68 331 48007 86893 B9 640 16234 5680G 02 176 96397 40348 41 134 42 742 45240 41 941 96382 71 961 98 145 77 557 80993 52079 31 249 87 637 32825 51 981 47677 20971 66281 78542 81 333 81 594 61 613 00397 86864 69979 93278 68 107 62535 93584 II 303 44 035 17 395 87 648 81 719 01304 87083 47 143 95 719 28404 50949 70774 28296 06571 32 270 37 143 84 827 64 710 91 976 39527 50654 26269 877'i9 31 003 44999 89435 20 151 69861 31 010 97 790 05 335 59 381 02 295 35 584 04220 94938 62 290 90429 00682 43 44 45 08 R96 48581 23307 02 591 24 fi74 17 119 129G9 71 539 36870 04401 86304 81997 64464 ] 2 272 27 398 39094 88695 25016 74 852 05 455 52 527 56 127 09973 32 307 10518 83 389 91 870 27 124 95375 20 714 04618 3260428466 55781 48949 51908 74 287 07082 42836 74819 58303 73515 61222 85496 45875 74219 47 324 7S 237 49 139 08 789 23821 05533 77433 53075 43800 73407 41 994 25 298 20 539 GI 427 58021 19255 33440 57 546 21 GI5 87 374 76 150 67018 05871 53295 97 553 60475 68 795 76514 72 306 13980 75251 85046 09191. 78 142 29822 90400 60561 57560 21069 64049 62 605 62 047 06 441 88 156 99538 52 139 53 783 71 839 09351 35441 31880 37548 73043 94624 61 171 00387 59579 77938 91936 80814 36040 88461 15 020 01 848 64278 68 476 41361 93823 07 706 31223 94 119 77 762 83483 94 541 72893 65344 31853 08007 43860 93 174 71 148 62 327 81 604 85644 65584 40030 15 501 03 856 64691 04713 61 212 92 301 06410 31 024 5i 748 90324 23356 09994 76938 58044 64659 82 760 43 178 17813 08420 01840 20791 47055 37408 55507 67415 38452 45449 7283493972 43643 18423 18880 47 277 49698 37 438 29578 54 552 19202 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 42 785 136Gl 58873 10591 40510 07893 ]3628 51 215 90290 62269 50263 90212 58391 12607 17646 29901 6841-1 82 77420288 55210 29773 B1757 05686 73 156 23 G2l 94049 91345 24 170 69 777 12830 72869 22970 52 166 96 131 85 261 11 711 77 586 31417 34 072 09035 18072 J9814 92983 10507 63 147 51926 28834 73 852 10 123 34313 71602 56271 21815 64 G38 85 794 96207 59 175 05 128 13499 64421 64721 34 137 70091 91622 65861 92 937 10086 39250 85 902 74 296 63 64 65 66 67 6B 69 70 71 72 73 74 75 76 77 78 79 80 07638 77929 03061 60528 83441 07954 83596 35655 06958 JO 850 62746 99599 39820 98952 43622 44 156 20695 09719 06319 80814 66994 06455 50498 19362 73 167 ( Continued) 20 IS : 4905 · 1968 Col No. Row ~o. 1-5 6-10 11- ~,5 IG-20 21-25 26-30 31-35 36-40 41-45 46-50 81 82 83 8485 59580 06478 75569 38508 07341 23 793 30692 70668 94688 65443 95659 18288 27267 50264 13 192 91 307 68 434 4R 908 06 913 10455 12883 21 778 19523 67245 60584 53853 24 637 83080 16444 60790 03991 38555 17 546 32643 69572 06991 94688 15877 4S 197 16019 97 343 30976 59515 52670 47 377 41 377 38 736 12451 24334 18 157 10461 95 554 73 704 52861 68 777 19072 84473 54 745 42 672 14210 65027 38807 65 122 35583 07500 36066 74384 38992 36 151 57 178 93 716 32 886 92052 95819 39510 78800 48 763 16 127 27 437 72 294 24 210 13 622 24 591 78 601 33 712 61 184 36961 59 659 16 563 37 992 94850 89342 22815 99073 65762 16894 59 780 46215 06831 35905 06494 61 773 12202 20717 47619 8883.5 90822 56196 49 632 07 ·177 36699 62 126 35 700 11 883 91 342 04285 31 649 86 283 79 246 45 134 58838 52623 07 759 27493 11161 66083 08 355 55 121 00911 14060 14845 41 839 39685 74416 53 152 54486 97022 80091 24041 44606 53 728 98408 04 754 09528 37821 01 392 42096 68 258 86 686 26529 73859 07992 51 777 70939 78576 24653 60860 292B1 93936 40619 46672 55 382 23309 53 166 67433 23 768 17 719 82 067 08 337 17 985 28825 12843 83824 63011 88 325 17974· 63281 69572 76463 26760 4936412369 97 377 85 130 45819 06 156 04 207 63400 65676 49 911 35 793 82 590 52 692 98901 80851 15077 02023 13798 3·i- 222 83637 73 331 18601 27585 32552 52979 04 III 95 954 05462 96299 97 34-1 28976 09815 54 130 14974 43 667 90712 08816 16435 266:>5 41 326 08 408 49 953 69200 90836 30 358 66252 93 146 55 160 40 34470883 26769 47 4~9 91529 90802 44 344- R6 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104105 106 107 108 109 110 9624-0 43642 03 742 83873 51 972 37867 5484-6 54759 65130 0486~ 846CJ9 58 232 88618 19161 29735 59076 76355 29549 61958 17267 10061 35203 35663 47 07 93 69 762 936 779 616 71 299 27 934 80863 33 56... 23853 5tJ9\)) OJ 514 63 780 24122 66591 27699 61196 30231 92962 30532 21 704 10274 03788 97 599 75867 48228 63379 85 783 60365 83799 32960 19322 11 220 94653 42402 07405 53845 94747 77 100 90899 75754- 709103 7803:l 70207 68829 55936 66483 33374 87539 03823 52972 16818 60311 III 112 113 114 115 116 117 118 119 120 35075 33949 42 614 29297 01918 28316 98953 73231 56623 3444:l 34994 41 374- 70071 14 736 09958 18065 36409 83232 99385 41600 11 133 07 586 15 917 06253 57620 52606 66497 68646 78 138 66559 19 6~O 99413 07399 37 408 48 509 23929 27482 45476 8524-1 35 159 68 980 14454 07481 27 499 35902 05 339 04504 83828 98 748 91 386 15470 2009473788 60530 44372 48355 98977 06533 45 128 15486 88651 74- 843 28597 74022 65741 22 596 93413 20405 84617 14014 03 152 22 109 94205 82037 87481 19 121 78508 20380 10263 37220 31 751 57 260 88492 99382 30934 47 744 22888 48893 78212 16993 ( Con~inlUd ) 21 IS I 4905 -1968 Col No. Row No. J21 122 123 124 125 1-5 ~--_ 6-10 ............ "-......-_",, II-IS 16-20 21-25 26-30 31-35 36-40 41-45 46-50 41849 46352 11 087 52701 57 275 20857 15633 92 694 77613 38688 25 163 65 251 36815 64397 04515 83761 14387 51321 72 472 05 466 39528 81616 07586 90767 40 188 34414 63439 67049 79495 91 704 94015 74 108 62880 11748 17 944 66067 54244 30945 69 170 08 345 84547 33049 96294 08337 36898 73 156 84924 48297 19019 32486 01 889 07 629 43625 11 692 25624 60873 06345 92 246 00008 55 306 46850 69248 14013 56303 81304 70284 90415 39904 88 152 45 134 70014 37 239 18637 05 327 95096 43253 80854 80088 80 890 93 128 52326 93460 31 792 87 315 48 585 24326 93614 02115 00080 63545 15021 33 295 37 509 82 IG:! 67946 84 14.5 09279 77074 18002 18 '_64 34677 45305 59747 16520 68652 79 375 33521 59589 20554 59404 41290 05870 82444 20247 48460 60833 43529 88722 94813 i4457 48545 75 122 92904 69902 94972 ]9 152 3602494458 54158 75051 59820 25704 22304 ] 7 710 25852 46 780 59849 47670 94 ~i04 08 105 58300 07 521 67277 69676 27 376 95 220 26665 49067 91 409 72 059 67 312 01 119 99005 81759 74910 613113 76503 11654 92852 01 159 55823 66821 96277 43947 64345 31855 34513 99893 55866 63267 47641 41 575 48257 51 680 15957 26340 73 701 25332 18 782 41349 74761 49431 83 436 11 834 19325 14413 39663 02 181 88448 10622 86225 49 767 50816 43 852 68971 18477 14707 83745 16930 20368 41196 66919 35 352 79982 81549 70951 77 54468 161 03584 48 391 31 70404037 97616 59 693 11403 65622 93997 22567 33361 07 126 37480 31678 54 131 68416 126 127 128 129 130 131 132 133 134 135 71857 92 784 04 921 45 197 85558 15 J91 25 9~3 063]8 56736 31900 90561 35247 11 72;~ 13 141 63 742 11598 00023 00867 74284 34243 93029 96 163 91035 90314 59 G21 01291 38384 66 164 54 155 72848 18 G19 74627 32392 7846462095 136 137 138 139 140 141 142 143 ]44 145 146 147 148 149 150 151 152 153 154 155 72 484 82 474 25 593 18 711 53 342 44 276 16 120 82641 22820 04235 13 574 17 200 28 193 29 593 88 627 82 157 75 363 09070 04 146 30 552 46874. 88222 87 873 12 102 05 600 42 792 91 030 57 589 37 403 88 975 86887 44989 93 399 52 162 04 737 32 444 88570 95 160 80580 60 478 95 C43 45 547 31 732 86995 35 841 55087 16822 45547 90286 21031 13674 18611 19241 73 707 583PJ 15997 19763 61 199 67940 22501 ]B 627 9087'2 36787 00441 58997 3262465961 20288 59362 99782 41145 48968 39283 73950 49856 37006 87417 94746 12459 14713 68691 73488 34060 9.5 93B 93478 42820 38603 04149 79 5.~:l 99326 22 186 17 198 49580 91 314 71 181 12 302 80 783 7637841605 05041 49807 46978 35482 47 665 64382 25843 14 750 17 334 07850 396]8 59481 21647 25366 05511 21476 482i7 7..! 015 59 221 41 867 03 343 52 680 70818 57 260 90 307 85 771 64654 01755 72877 06554 58905 57216 56487 96 169 07 654 71 803 59987 09971 61 459 46376 26825 87 112 156 157 158 159 IGO ( Continued J 22 IS : 4905· 1968 "'-" '\.." No. Row',. No. -, 161 162 163 164- Col 1-5 6-10 11-15 16-20 21-25 2G-30 31-35 36-40 41-43 46-50 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 27 767 13025 80217 10875 54 127 60311 49739 78626 66692 44- 071 41468 94559 41615 50273 41396 25607 06 170 60808 80940 43584 14- 338 36292 62004 57 326 42 824 71 484 51594 13 986 28091 85 149 37 559 70360 93 113 80504 85301 88977 29 +90 69714 73035 41 207 74- 699 09310 54066 98525 90391 26629 37 301 92 003 16453 99 837 07 362 49554 49678 64114 41794 90670 152+3 24335 61105 19087 47 i24 66733 47 ·~31 43905 31048 56699 2-1- 432 24896 43277 58874- 11466 16082 57411 06368 53 B56 30743 03670 81741 24472 8877O 71529 7H 920 7'1. 6~2 979G5 88302 98041 21443 54444 74412 81105 01 176 44893 10408 36222 80582 19516 90 120 ·Hj 759 71 6~3 13 177 54480 88 94·0 00 307 98932 63 57477 756 69457 40455 03 577 31 370 61 826 70 495 33 230 91050 67011 97380 61 764 11 788 50669 86 124 23604 15995 11897 78284 3138411657 91339 99396 57 649 28 977 18555 '32 350 21 529 13058 06651 10404 97 586 68 224 48 139 51 247 23554 69321 92674 46347 51 92·~ 13897 22502 63680 63 266 23 896 64937 02985 53424 16218 16136 55452 54716 23 417 36732 44302 21785 47458 4~ 40.5 71209 855lil 07 385 90726 57 166 9888441808 68984 83620 89747 28838 36421 IG 4g9 18059 7194t- 92633 40333 6705155292 21036 82808 77 501 41 101 91 173 64809 98 189 63032 96717 92061 3941-3 29 G71 513 137 181 182 183 184 185 49386 06 312 60 94·2 92 329 77 936 38 101 39641 84054 47 468 43321 10 ]74- 29420 so 433 81831 296.51 84- 215 54241- 10701 +1393 93 136 25722 0356117820 22 751 36518 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 92613 67667 24700 76479 95889 570G7 47 6·~S 13885 70669 93406 89 71~) 11947 56203 19324- 2050.... 60631 69 181 96845 38525 11600 71 594 14004- 23 153 69249 0574:7 69562 62342 07589 08899 05985 64 281 66847 72461 21032 95362 49712 58275 8951415472 12 120 13 173 86716 92581 12470 01016 34030 50259 73 959 4687460883 33365 38746 02262 56500 78851 26313 78438 15292 00857 55018 16499 77463 66276 76 139 56374 10271 46092 40277 09819 36 786 8706f 55 387 18396 59526 35824 36633 26787 11049 58869 49 226 13075 72681 73538 52 113 71 703 68424 6093) 72 o-s 33 220 77 837 60726 75211 46345 87 19.5 76 145 30342 37088 7346j 52 109 21437 ( Continued) 23 IS : 4905 · 1968 Col No. Row 1-5 6-10 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46-50 No. 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 19612 78430 39 141 77400 64 756 80457 9290151878 03551 90070 98 884 27 369 59066 91 647 83605 24895 35 720 14 141 27416 82071 66 209 86882 75974 93 783 92419 88530 26 556 53410 75670 07429 11 661 94770 28000 64238 08 747 12836 5644122998 09483 94050 06 830 53473 6333.c; 64 169 39542 70 774 95 596 38649 92176 81 007 53 656 07 541 20483 49 022 07772 35439 20 094 06343 72535 47 749 77603 73258 03469 29718 45938 14 663 53633 43514 98 588 71568 46758 73 750 57 256 93119 40 744 65669 71 794 50678 38447 18 135 86868 31 340 03274 06453 36908 12 665 30012 26256 66453 43423 66677 2531107565 43 321 11 073 04 909 12822 26967 59 068 89759 92675 01 703 75318 18244 88821 75989 37016 82556 53771 51 803 56 346 71 430 70863 03 748 37481 58278 09 495 49 829 7567335185 70472 85 788 61 342 56077 56974 70207 34 264 72 709 06886 23336 19 818 05 707 19360 49088 49325 43951 38 8-31 04838 4490174291 91 623 46 833 90379 92344 53841 61 275 65 248 37 562 3137410536 21 445 82 793 24831 93241 14 199 76268 70883 68 002 03829 17443 72 513 76400 52225 92 348 62 308 98481 29 744 33 165 33 141 61 020 71479 4502776160 57411 13780 1363252308 7776288874 33697 83210514660908850395267430530621706700019943980767 68 749 95 148 94897 78636 96750 09024 94538 91 143 96693 61 886 05'184 13651 00566 50 958 57621 09282 23394 05280 95491 78 521 96345 77963 07 520 38 423 02 463 15880 71 926 64425 79782 35337 75763 62546 21 220 17 695 64547 25844 94206 37470 97976 00 104 44579 31 151 11 294 02 309 65 533 92261 00819 28 108 23924 74538 47075 96892 00292 58 072 46850 79 139 93432 93622 38306 18248 85932 32 364 23238 70 703 21 199 17 292 59 144 16554 49440 44 553 88158 25240 24069 68 990 37981 78435 37836 04345 32 192 75583 44 053 91 691 01 748 85 736 60 555 88 190 00 224 16016 30432 64672 0531353439 47 511 58483 25072 29519 60 329 95 955 38527 09037 35428 94919 15092 82 639 90326 75 704 47 357 41 690 46 148 33 928 61 781 30570 00042 81 077 90960 43561 26846 19510 54624 50785 20840 40338 67 328 14 258 01 817 48898 90 194 83229 31 543 41 849 14908 87 342 52548 71 586 64756 69799 02555 18094 72434 54034 86583 23435 54814 29 236 07 396 92 525 18329 10333 95216 93865 08830 78818 54091 63 417 03324 63 314 74410 16613 92606 66251 83944 24065 37 777 12 152 89 215 21 283 06999 36 168 64865 44608 60096 07855 21 282 35 947 04986 12 991 94915 78234 23 191 35 774 52456 08458 10057 05 088 30 722 88581 26857 65617 13658 93 176 21551 39269 21 296 67 807 83666 93516 48 199 50001 74693 14692 73 766 95366 42 332 65 825 22 102 60098 19238 9483451 081 34851 ( Continued) 24 IS : 4905· 1968 Col No. Row No. 241 1-5 6-10 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46-,)0 -----------_.. 05249 29329 5646399380 96 296 33 121 98380 36269 52567 64350 19715 94082 3879385774 54 196 34 108 60014 07 201 16315 53969 14738 19056 75 814 62 448 80395 _--~-------- 242 243 244 245 86667 43 708 13939 460G2 85 986 71 171 46385 42 175 81114 54358 66354 9:J 692 2764766146 ]5 102 28 992 88350 49 182 64578 47269 25 527 63210 63 165 49 126 15747 246 247 248 249 250 78498 90830 25955 99236 49553 24241 08 ]50 89535 32 151 07075 83 155 10252 11 314 50363 26860 27 799 12 364 71 210 87 052 50 24-1 5946758309 73452 17619 27635 56293 86040 02596 52 403 94 255 49465 94365 34261 37 110 83950 61 630 01 929 41 659 32031 90043 79418 85447 86219 71712 29776 81488 51 667 99004 68 656 38074 01 020 86 379 48498 41 800 63026 46581 92 560 08 769 66538 45424 97966 17 165 39098 3960B 93478 14322 61 079 81 115 88559 63075 17340 37589 70322 66492 74083 80680 74508 09938 95363 96712 87834 56421 91 700 01 655 26 351 61 499 12 363 91 830 39318 72453 76537 12 037 23982 75992 58044 91065 96910 49625 92476 13 270 74154 87 147 60832 35 933 09337 59328 33 579 44 420 54 142 79883 43 286 91 064 99969 95 144 08 703 91 041 77 323 81 079 73 100 88618 23891 87418 49416 83534 19 187 08059 90 785 97 889 81 399 58 J 30 33689 01259 62486 41 701 93223 41 682 89604 73020 69853 00222 54577 74821 90 183 85 057 72 310 75036 28850 82827 64733 07479 71 750 77979 50525 883iO 86602 84338 24855 58417 10317 89792 06325 56534 95 495 36871 45 332 23 085 58875 60564 69549 88 620 37639 08 782 94088 67971 55546 54477 87 313 50274 10659 34 114 13684 87962 28778 62604 79794 72615 95 304 59 DIS 27284 92 824 02012 12 505 91 088 92 748 65960 62006 14558 29693 13 764 45 191 83987 40859 52096 68433 11787 23006 64 424 77 377 45 127 93686 45417 20268 76677 02 110 64439 05 614 56320 45026 61 517 47 335 34 963 67691 57 791 58 167 89985 22458 94984 17 315 :~O 214 45316 46265 251 252 253 254 255 256 257 258 259 260 261 5721337510 40725 23439 04391 67 317 09918 45 161 46 527 68 224 04844 30246 23 313 15626 19444 46467 74558 29899 73445 06949 07841 72906 48799 32903 84758 02 086 18 256 44324 68219 30942 16250 71 594 01 317 31 176 36130 07 361 89485 68009 49560 47 505 51 207 08582 83 462 67 762 71 328 01 305 36936 35 148 37 782 72893 19769 38551 94626 02963 41 482 83879 44942 63915 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 00964 66715 70595 16644 31 036 71577 51 091 70 102 72964 84 906 4280127917 24743 48023 76636 56907 52 293 47 953 07965 65047 08 712 77 114 13484 17482 39225 48 190 92 955 52 319 32 705 52 653 62234 13124 76471 35889 13255 04925 46288 36788 93 196 50 009 83 464 28 608 21 769 30398 44 855 6601 I 25 242 52500 46 174 22 684 09376 16 322 95 795 64 130 31 661 31 287 03 054 88591 40 954 36693 59822 ( ContinUld ) 25 IS: 4905· 1968 Col No. Row No. 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 1-5 6-10 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46-50 88298 07839 73298 12829 76569 41 665 58652 13 607 55715 04 110 15489 62 735 51 108 70474 61 072 41 339 49983 00657 26203 66683 16030 99218 48 717 00838 48568 62 106 01 669 76 173 65933 99001 42480 25 624 92 926 50 385 36491 44203 27464 43 357 51087 09 796 15 372 02 547 75 705 91 711 22 587 06732 79553 77 334 98234 47 349 38 781 27445 89787 80370 44 363 71 995 69 187 96 114 56504 39592 77438 55 749 99902 56857 61 546 67999 18 087 02 906 63563 81 970 69019 92 498 52 056 26624 01 507 06 120 88044 14 751 46864 72 437 78 079 03930 77861 92 581 67 757 54 116 14 186 63 740 43561 59022 99721 54606 06 740 22 667 28556 90287 30682 93730 85564 46 023 91 161 32 322 37 749 80906 90 181 32 231 38052 89863 89148 71 262 73985 49 737 04 786 50463 40223 15 378 16673 92 380 54717 75075 28 970 38984 09541 05471 58019 47014 76 877 40 158 15673 31 752 36912 14761 08368 20466 29695 33477 00001 45488 58977 75 768 10 192 15504 96 305 0901B 37 348 76869 17 529 44651 82581 14479 19453 64599 11 585 77361 09 171 56559 66 102 30434 97038 18 570 53 859 35 151 85 739 64337 58872 18285 87 867 72 703 28956 04370 28267 05 134 80051 82 948 80179 48865 67814 60 23~ 12 247 59877 76 III 79840 52855 25988 24 140 53860 4062545545 65003 66524 23965 61 115 62 155 58752 49499 19355 19 124 25867 37882 38 192 84 570 55 176 25 367 71 724 34857 57319 50 445 44 301 95261 64 098 16495 46 138 54334 64 894 61135 23 373 01 587 79458 19958 50 332 31 300 08681 26225 97 534 07 158 82 763 7125125572 29991 96526 83642 69 167 86018 44 114 53805 52 065 37632 82 576 69023 98949 96526 33 692 50 335 88321 90303 50486 27 882 45 144 81 020 66831 74058 30348 59557 67098 60774 21 057 30235 29406 06026 64 ].1)0 08 853 80274 82 805 64971 03606 28 749 72 111 09 513 85062 62 469 II 885 16269 63213 17 882 47427 18293 70 174 23362 09899 66371 58068 44 115 37 044 07494 25072 38478 7977193328 02820 91 659 02 677 06767 75415 79553 70915 30] 04 35 240 94031 99321 78236 56592 60 958 44 346 57345 82 517 23085 64483 49666 74973 76033 09 963 11 468 13 746 25 775 69 442 09367 66323 22 038 55091 63 127 79 937 32960 12 779 07 521 78985 37871 96 848 26 439 66231 43035 41 712 11 273 27441 74531 31197 35278 25994 82244 00332 20 385 40064 77879 85 778 35345 57 782 75291 6692754069 12818 96356 38097 78 294 27056 95385 63 695 66 913 74859 90879 95909 29212 72905 17 893 55 293 15409 36850 80692 02 680 86989 94994 59817 13062 25343 42093 36636 14486 16 100 14916 26906 41 212 41 288 53 200 07 359 24 109 43897 57369 70 198 40 993 06 449 03451 15592 48 492 01 616 29884 24927 '20064 83094 22317 24671 57 594 91 330 ( Continued) 26 IS : 4905· 1968 Col No. Row No. 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 1-5 6·10 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46-50 60081 92936 72 302 75064 80458 69902 34 870 88 684 53 844 96 326 65 728 48 563 69841 29451 36 170 21 529 37771310021831193285 27 286 49944 87 693 94282 26917 94 267 73432 31 102 41 429 21406 71512 52 028 18 638 73287 73332 11 354 31 312 69921 79888 06256 24 ~61 19278 58 124 12025 50326 38 422 10371 82 119 92261 08582 65206 01 419 60 982 26201 31254 55585 57 741 15 795 58982 88 526 61 638 05756 46 064 31 926 35 145 73 047 57213 96946 45263 10785 37768 07 215 28 151 36 174 17288 19646 85 108 04020 22466 18655 32 121 13790 42213'" 40450 32902 57411 77 327 87082 87 133 92 124 52 110 18807 96363 04538 33 159 57 580 51875 39 133 24541 50859 24 200 53300 65919 01 172 36233 94325 55067 98 885 ]4 106 59809 99246 21 615 67 672 98844 00865 85 727 52 987 05 750 19 384 33684 78859 49 762 40801 86291 18 194 90366 82639 26 027 52 692 62 406 76 294 41 848 63 0 I0 16525 64 326 22 086 24469 57407 96033 31948 14331 58335 1597780336 81667 95 270 53445 77 385 23854 72 119 46 180 33342 09 605 58407 95094 76990 01965 65 3.19 32 131 29441 91 723 07099 57888 27 697 13706 54235 66605 03 085 16 14] 47951 20632 16 124 84470 58256 65957 89638 91 933 49591 39260 79656 29255 95803 61608 28299 07 056 31 441 70355 65 379 80475 20674 53 031 94 366 0793710054 70 113 90 563 92845 20 714 42 301 62460 97 802 72961 57908 48420 92551 71986 69621 23 639 64 174 16570 21 124 33888 45 605 21 140 98261 02 727 97712 86 637 27 898 75252 85 349 72 049 80847 46229 AI 083 03819 55 926 78835 1223707866 14721 20990 11 669 21 738 76 431 25 566 54 102 57 041 2506151612 37 564 24365 86816 48353 75 247 32 660 27 684 63027 60666 00264 78812 10313 48437 46456 35771 02 838 35 487 97198 97 146 97 510 20 129 19594 39890 15 328 74931 98 496 12 734 13767 65654 16849 77 III 5886.5 41 207 1102163813 05345 69827 30 849 03591 43330 60597 14966 44 887 93507 72 706 56239 08 537 63 986 90024 79213 69315 35303 26222 69561 36396 65099 78416 69 108 48 181 03937 34988 55402 35290 01 145 53 743 71 126 60775 43816 19646 46460 46 065 85 700 52 777 46792 54563 96004 59952 27896 5069164709 99859 77 644 25793 34683 12 147 91 037 23 180. 43 164 92 740 46524 10 362 39892 14213 81 419 58 ]58 44797 68 124 52 643 95319 96627 99975 73627 55 2tQ.. 52002 30'094 .45958 79327 31 962 10676 12083 40986 74 129 42837 70447 60934 08512 70997 77 989 60660 42081 35045 53444 95030 53 127 18414 18991 21 913 79 332 28982 08816 86602 96 758 71231 10053 50483 13422 51 479 61 882 5666776783 64230 84418 38640 20 383 9772379390 56763 12868 54488 20719 51 449 73637 20405 34690 08697 14661 68266 52 935 02 293 66 169 ( Continued) 27 IS: 4905 - 1968 Col No. Row 1-5 6-10 11..15 16-20 21-25 26-30 31-35 36-40 41-45 46-50 No. 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 46 326 67053 52 928 73 797 21 407 -11 064 34206 32 205 92 748 02 754 44968 37 978 68318 64405 28 136 33 BOI 55 106 38 543 15280 35263 89 109 96352 77 816 27 226 54214 47 599 98 126 04209 53640 13266 57 477 47 394 04075 76783 06812 93415 69602 20 225 60 171 20 679 39923 03949 70244 49494 90038 01 790 63 132 49 508 84 147 41 731 II 397 73873 22 312 90020 19328 82087 50343 16 585 13 558 53 710 72849 14 106 47 204 92847 64 175 94472 12 158 53515 95 326 34 140 03 941 39475 66644 99044 56633 23464 46961 92 525 29603 56 304 60625 70082 91 954 41 878 7263B 58817 38837 98 425 79 835 37 068 92 973 33475 78 792 07 492 38 161 86091 70519 86841 95916 16667 22 711 32 938 34876 85572 43 701 64 150 52 095 64 342 93381 27253 39007 90621 87 803 20851 50612 43947 66567 71 179 42 535 70 043 28 386 02 303 17 401 76635 69692 86400 40210 02 451 94 867 32 753 50014 09 720 87 508 52 413 57475 59969 14567 73 742 89 759 79008 65547 28083 45963 15308 86845 87 753 64 833 21 223 37 113 02 734 14619 23820 97 815 84981 55289 43 728 19359 04 859 86365 87 568 22919 48642 92 693 83227 51 599 66213 96346 35 423 41 224 91 059 92997 97 852 88466 95 III 13771 90398 36780 08 766 76686 11 231 34542 18633 79262 80688 15569 68 652 00492 16662 80751 99070 38320 29344 99552 59052 04671 74284 84676 91 208 93905 21 386 19415 38429 98 342 76618 B6413 92 599 16967 56 072 61 794 13 407 80968 98449 72976 34455 63562 56870 55450 39 252 76467 29397 26686 80286 90 181 05 761 50687 60726 35817 43265 76469 60077 93449 94859 78666 77 178 84419 67 177 63918 60 430 28237 59 049 75 766 94053 21 451 11 946 32 019 70905 81 619 36 810 35 066 05 607 93 761 48 722 47099 86 311 84207 55 756 19606 75 678 67 147 67 136 81 678 65 507 84041 82 638 52 679 26213 55 571 68219 96677 57988 31 336 91 604 03942 72 109 94072 57 105 40650 05 239 45 317 78 353 43668 38 770 84988 13451 64 856 78 324 96503 30 332 82 220 69 384 57 598 84977 84 126 60892 94 121 78841 83 749 75 379 63 110 18601 87 765 46 197 03 745 13464 87 696 17 320 91 300 03529 43054 93211 25 279 14033 08 175 76475 94446 31 059 12 923 61 458 60 718 48 696 31 626 68 137 80487 87 591 74730 54326 14881 85 998 66 684 53 329 74703 11 271 38595 01 074 18901 98450 67 562 78 760 94008 64502 77654 52 722 90041 23683 08285 64 553 97812 43636 22937 92795 97030 41 868 22 345 21 619 80603 10642 15974 24089 82001 77 379 70301 68 441 33807 78 381 41 531 43659 89405 49 358 20 096 95909 07 768 11 728 85 12460274 52 768 54 165 64 765 37 279 08037 90228 23664 75476 30780 81 015 97444 81 285 37 507 68417 31 954 83 795 26245 14816 19983 73017 32 188 13803 79732 68 852 41 679 55 309 ( ContinUld ) 28 IS : 4905 · 1968 Col No. Row 1-5 6-10 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46-50 No 401 402 403 23780 45325 88 240 42 789 97 523 59083 79932 21 149 45055 46497 81 905 91884 25728 89 ]78 30935 49789 54558 56631 83558 74321 44047 52372 24902 98377 53629 49867 16719 46970 35747 71838 34534 17082 84721 16908 86370 64677 95812 09199 44757 63 168 28391 05490 92 457 69 758 17 264 95 137 44236 03425 95091 28626 05940 65974 89 200 79 70 I 82 840 76538 10089 1759108367 H7297 55583 8125G 59418 97 521 111B6 15357 035GB 00450 94 (J96 II 370 91 157 48487 29511 5596H 41472 89474 59556 37 119 30985 48866 32846 70 761 90 115 96644 58976 36211 59501 56983 89795 84344 80517 07485 60605 95719 70417 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 44373 31427 LA 381 3656B 44155 67286 57 R97 28262 04052 00919 86207 65670 44285 06 903 20 834- 49 701 95 735 14262 32252 68 540 39427 44 026 47 257 57 375 41 .162 83883 27 715 10 122 67 745 39483 11 385 63 292 92 305 78 683 06 146 51 094 66507 74 351 75376 45886 67 635 07757 25 137 11 551 14288 06665 75901 65 737 65202 43213 88933 9037438785 7566462831 15 038 38338 5) 206 65 749 34 119 71 516 74068 66762 11 428 70908 21506 00480 94 183 78484 52 539 8680G 69 944 65 036 27 BH2 02 530 04918 08791393429496322581569171754183578 79270 91 986 99286 45236 44720 H1915 70881 97081 18395 88862 31 960 67351 16075 20517 69980 25 310 73375622515887109870 30487 38794 36079 32 712 69473 45950 18225 09 U71 27703 83717 18913 42470 26995 04931 15970 15 365 30260 31 776 38332 74547 41195 60714 50933 73622 37681 41422 97303 54919 64452 19704 58 135 91 953 70538 11 130 88502 08816 50003 86478 22 236 16630 62 789 01 759 48936 55451 75 179 37627 419 420 421 422 423 424 425 426 427 428 429 430 67612 72 738 59 Oi~2 37 595 59609 35653 35 354 65 770 79452 71 674 89294 06 144 45907 78956 07 526 70 169 26997 23 5+4 82 841 91 949 59232 82041 99238 II 478 07985 637.1A 68 387 69 365 71 356 064-87 43693 52661 55852 40997 86374 40 044 13005 05245 65550 19290 70 152 75300 00343 66449 22501 50486 OS 407 58 779 '25 193 19 792 99851 05438 01096 72269 31 B'l() 99287 59928 25503 7'20:35 26542 54600 79 172 63856 33037 45753 60 159 8B627 75 790 38454 96 110 39237 431 432 433 434 435 436 437 438 439 440 24805 63215 38 175 38784 32295 10894 21805 65245 88548 65626 75517 69737 24060 40285 19 195 80281 19017 3384·6 77869 14321 38855 24 B2G 50917 25 147 85407 55626 89322 95 102 37926 69214 5217521697 15232 70043 87073 71 467 38579 19453 60691 31305 80358 52629 79419 22 359 87867 48 296 50141 46 807 82 184 84665 74511 59914 04 146 90417 58 508 62875 17630 21868 30322 33352 43374 25473 04 119 63086 14 147 14863 38020 986285791622199118654291162651782900939277294 21043 17409 13786 27475 75979 89668 43596 74316 84.f89 ( Continued) 29 IS : 4905 · 1968 Col No. Row No. 441 1-5 6-10 I I-IS 16-20 21..25 26-30 31..35 36-40 41-45 46-50 ----,-----------------------54941 48575 27 119 18570 36050 39 B29 98761 97479 91281 74,396 42 326 93 929 83585 27 548 3l 982 67 126 00818 62209 11 331 71 732 42 467 20 706 60430 41 246 33 150 49602 40364 17 578 42 096 40 076 85857 93553 82514 91309 5457453266 06 779 41 957 96 837 74839 95992 77822 16060 72803 73736 15564 309B7 79 172 57 R75 57 146 06974 01 836 00414 37 516 56 455 76656 09 136 43740 0683:3 04 704 88 801 31 929 59627 98416 27 368 74391 90913 12 830 34 934 20 292 45 1~45 41 059 55 142 15 214 42 903 16799 88 254 95984 21067 30302 70040 13351 04716 57657 72 764 45228 57238 95327 91385 48321 35 352. 96 779 12849 38 III 96436 96 263 28357 51 718 39 564 97 090 17 368 65636 23 797 07 598 56 188 72903 99 937 78473 85 999 21 584 46 379 63079 50 026 21 060 442 443 444 445 446 4,t7 44B 449 450 451 452 453 4J4 4.t>5 456 457 458 459 460 461 462 463 464 465 466 467 14594 33398 66446 49211 G1665 31 159 61 36 62 24 53 063 590 851 343 129 97 640 7505:l 48 787 63046 77 693 28492 48 4'~2 76895 77063 57 343 01056 18730 07 794 48883 70 171 55029 64463 95934 52553 61197 22363 97 639 65937 17802 31 535 42 767 63053 25 926 20944 19306 81 727 02695 78864- 12698 15812 97 209 38827 91 016 69755 99224 43999 62 879 08879 80015 06980 79069 37 409 7503" 69977 85919 13433 89475 28447 02 081 25022 43 108 91058 98 172 12523 66682 27 534 96197 60475 77 15459431 10371 50058 09 132 47 508 92528 15 ·~37 21702 20378 555346·l736 Y2590 31 .5R3 45 570 44500 81619 22 101 76713 09806 14534 91311 6.164B 57033 19510 99375 32 278 10443 04 155 68970 89 745 79004 47592 45427 94015 34018 16658 84 '~39 73 731 DB 03R 26648 82635 99 L~19 93895 84 493 35 356 08 996 82 248 07 235 76428 07 493 08440 38840 22 362 99012 70236 12 307 29 fi33 44 760 34 165 42226 97 134 68 739 30086 98001 07432 84816 05 195 66 '~92 70055 10436 25824 02 180 89654 60 347 85 665 68400 82 478 HO 683 83011 13611 56264 44659 04056 34530 35459 96 056 78 333 34306 94546 46 755 80440 34009 58981 77 751 67202 66362 18287 14370 20027 77 797 44148 62 398 67207 72 577 48452 07977 30382 75895 8453407959 39 085 29466 15827 45430 11 449 30461 32 141 16591 73743 36455 33280 82310 8167408405 17 94428 ISO 03 113 76597 95790 39963 23090 55263 47878 38312 98335 26 567 48 151 20 121 16288 88824- 29 347 01 952 54102 03818 61 384 91 280 57 422 26471 08 669 53375 48830 73 151 06571 76 609 32 138 468 469 470 471 472 473 474 475 94261 78 191 25287 71 561 9547C 23 123 53688 95675 13033 56 906 01 143 63 30R 49864 54478 48 238 05 998 04855 27029 01542 72443 65434 12 124 91087 87800 86800 16781 65977 65946 51 233 81409 46773 69 135 92933 77 341 20839 36 126 36 146 62 663 93235 06 283 00 740 78047 30 523 53 308 80 172 25 553 50607 47 881 22682 66 109 83 767 22 486 41 279 90722 92 592 35 900 476 477 478 479 480 ( ContinUld ) 30 IS : 4905 · 1968 Col No. Row No. 481 482 483 484 485 486 487 488 489 490 491 492 493 494 1-5 6-10 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46..50 44906 96988 75 172 264·01 10 157 26017 33 193 04403 30937 47391 57540 81026 71 179 39701 99914 09559 92710 32872 37217 15232 67885 52380 93 140 84663 91830 81746 88088 06671 24524 91496 31492 08 446 15618 49 794 91821 03697 45832 09021 85 101 60984 33 143 51158 55 187 17078 57946 65 651 43 294 05862 64946 73 165 13486 87 597 94372 30 (J65 27 719 37 795 11036 25460 63638 20033 07454 73 176 153]3 38765 35582 40 400 05 065 02 571 10 160 47 805 48855 22445 04446 32 775 00216 94880 74760 66819 75477 32202 47271 70660 82265 71584 99838 81 096 83 423 84908 30 460 57 198 31343 39 879 52 594 87 439 42 125 30579 16051 833C6 83 715 13648 56 652 01 184 40 725 33454 49 333 83 246 99 644 82 500 15544 77 589 25 546 83 769 62801 66525 99225 11 325 75307 35374 30068 22 348 38655 57425 78382 57696 39839 94279 86 206 67 295 32 399 83 145 31180 58 682 43277 86 354 89239 44 188 04475 14896 94737 35 250 34 755 49925 33 225 19 136 61 324 80 056 62 771 74200 31 962 16881 47 589 85937 50 775 53 558 96537 44979 12291 04035 42 33402 766 59513 50891 19681 30829 66971 02968 39 935 33 623 80 423 39750 00232 82 972 16530 04 546 28847 26532 13400 04219 69973 28323 63 992 635]9 56 643 57712 26572 75306 75986 :$6 170 540J5 13423 21012 38321 96 179 788HZ 59674 J 1 785 10465 48446 84503 75 147 79005 46836 28853 89 452 65 131 29073 385B3 gil 790 03496 41 863 44- 92·t 74642 66533 76 992 78428 87359 41- 272 (i7209 91110 29931 92408 11341 59 3.53 92 248 0897473315 76764 34899 35637 82 126 48941 49415 69 969 98679 75079 36681 87422 77 727 47803 48 100 17439 90217 59691 36277 46358 72850 8644-0 27 315 60449 25 581 18 70:3 65471 14573 62 509 12 799 47997 97967 7107417962 87 530 40889 26815 79398 09796 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 47388 85485 40646 31939 03843 9056489696 49618 99087 57 .170 96791 70 196 84413 09839 67228 12417 4719J 3B 906 34533 51838 30300 H9971 67923 39 118 89 939 05 769 16122 38 181 39956 28805 35634 25 702 85 787 02 529 88541 04297 73943 42895 47230 19 748 31 414 60 748 56498 42370 86 712 51 845 28 129 16293 61129 06 890 58 448 31 718 84257 08975 64947 74219 02479 43041 28] 74 40 852 54669 24823 39203 91 100 11 057 38 783 80 723 46073 87890 01 958 45 141 46292 80823 10891 37692 966H4 20,)35 28673 83450 28397 93 581 67953 99295 29 769 26307 89963 51842 29991 50 293 43 574 05 283 64 730 47734 91 252 17 941 50 492 18 107 89 186 08 311 16871 76771 76342 96960 75044 56585 60527 67 177 62561 23253 55060 56 929 03568 19707 18417 4-0052 65 333 54623 08420 48 527 38375 74 187 48 379 0838R 22 009 05 979 51 04-7 17997 08459 95699 38 722 38450 76330 94632 61 028 515 516 517 518 519 520 86570 29039 ( Continutd ) 31 IS : 4905~.1968 Col No. Row No. 1-5 6-10 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46-50 521 522 523 524 525 526 527 528 529 03 553 05 128 59866 II 774 86 746 89 698 7447387513 17f90 33049 0262241026 08095 64981 28 180 57 888 56 316 98 849 51 632 12874 77 884 64611 SO 393 61 477 84 746 09 110 38 688 56405 08915 89434 ] 3938 37 723 72 762 54 799 82 160 07 032 19736 58319 89 i31 28302 284,85 39 968 17839 11 467 38669 51 281 68 124, 56020 37 810 6142772914 80875 41293 38 629 76962 17007 88972 32517 16752 23 840 24729 I I 361 01804 84225 17477 55484 75 139 71 794 28 6R1 49810 30 883 69523 03562 78040 46831 65 779 45 405 50 186 51 745 59541 29 710 95 583 97910 84414 75 268 41 439 40574 70 786 00 589 06327 30246 3713768956 48297 70 340 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 38554 86836 02 195 30270 00234 21 424 26664 63804 59 767 52 497 24 227 83 152 27 973 68 568 68 465 98 500 67 202 85 199 27908 67022 01 671 25539 H5098 18421 13264 30 326 32 604 92 073 14 793 91 592 69125 74698 77451 61715 92 735 45 775 05 720 70254 18439 65919 97 043 55864 60045 65062 17 279 39 523 47 367 17 139 60768 67 255 53 362 46569 66519 29861 07 595 99826 11 694 57 622 82 691 88 799 28 119 45206 57 571 52829 00 134 64005 46 262 93 328 51 238 65621 56 786 48215 24541 00838 12 933 99 308 73 262 15442 12485 67 237 53 364 36534 21 398 18 369 77929 28 540 17423 14400 43350 24864 42 760 43 923 07 570 18 192 19657 54705 18 579 99456 24 279 96212 76 029 91 807 62 731 74323 47 275 90 066 67 397 58001 05985 93402 41 380 09833 89 215 96 335 81 153 03 774 90039 82 534 82892 20527 03 974 88 228 31 000 36826 58 705 19404 89819 44 993 93981 55567 57414 23 557 34240 77 241 05 709 7355378280 64603 59 752 07 7133 04351 43983 33356 37 850 66 128 66416 72 336 80567 59289 19690 72 353 12 979 44 365 49076 78 143 9559181168 44233 67602 47 350 21 234 96485 22121 45064 50924 68 590 92 754 50864 29522 13699 33834 98078 707B2 73306 67 440 02 043 13047 79 630 76417 30757 85685 76911 36619 42 541 91 844 9192725976 38615 31 303 16016 41 532 78004 06316 24204 58816 72 683 96883 51 479 04024 91 241 60090 74420 23281 84446 38361 82 797 10994 53814 91 654 64 Oil 69229 10073 56949 68887 34578 8907771690 28650 53 700 76508 23063 87896 99289 65 290 34392 37 149 05220 76428 92 434 67 171 85 347 85815 18580 90407 90551 03474 76025 35870 89 158 7388767928 45968 73667 67 622 54579 60 664 62 155 44969 82459 40873 67 547 09234 11 129 96218 41 590 44 638 54040 50563 36492 1193704074 76045 78 649 56441 20681 59 748 06364 89 772 24405 09657 01 887 35064 26921 18710 91 881 00545 38 159 16089 85909 09018 19252 466215257794853 53958 96537 95542 52 226 96618 32 831 94653 06 793 23 465 82394 09511 99271 80248 30 036 69729 05417 98 164 29009 17 173 82503 58953 ( Continued) 32 IS I 4905 · 1968 Col No. Row No. 561 562 563 564565 566 567 568 569 570 1-5 6-10 11-15 16-20 21-25 26-30 31-35 36-40 41-45 ~50 18532 689 52849 33298 81 569 ~6 10 721 39 755 31 652 87662 83651 23 790 18 370 88 318 00 157 30 635 22029 39547 79655 61 849 35 795 33842 97035 63242 65814 56 429 42 160 41 963 62 369 14 705 98079 48 524 00740 11 250 60923 40 168 47 778 36666 18463 68 685 33 501 00 881 07993 57 518 69411 01042 25579 91 227 12 618 73793 36 722 83 489 31 671 20 306 39266 58931 59049 92 754 30493 57 855 83 886 03 464 86304 25 301 80320 03892 62 394 37 760 11 569 81 830 96 828 80495 84· 358 10653 71 555 58701 57 462 12 970 01 374 34844 85 188 72435 01 411 73 723 06795 82 870 97 459 7340;l 51 208 89416 74 740 12457 07440 07448 13 183 90 954 85476 21 904 87 175 58830 12 625 89565 57905 85960 21 118 73216 73 370 03 874 55 080 93 418 52 384 53 359 09 581 56690 22 140 12 342 11 944 44 856 61 880 94 084 43 306 571 572 573 574 575 576 577 578 579 580 581 582 5B3 584 97614 17 033 76 624 3'5 660 18291 33 557 91 408 50 106 57 782 76 162 96584 67 097 29911 34784 45 668 93335 75608 58581 64013 47838 02 370 68 037 00 405 00 234 91 656 87 793 80220 10 099 63951 71 724 81907 95 523 65690 70950 03 459 86'853 39646 44 364 63562 46712 49067 82 985 51 270 82 856 65205 84284 21152 00351 79430 16 906 59452 57 453 87 807 25 384 75903 43635 43759 18173 49 181 90 040 4391146712 ]6 120 04454 36894 70850 56754 78315 99496 50048 83845 93661 85 114 69666 40 281 70986 97 252 85998 55 743 94697 90857 05 728 13 722 53 723 40028 04055 66 568 41 178 54680 29 870 15860 90871 57468 41388 10 143 68890 ]8572 86 B53 94 786 46726 46577 44024 63851 34 457 41 213 43301 23717 46 386 30 789 G3 632 71 048 15 581 39904 75774 77495 75994 47 712 70 355 16998 56005 05230 10093 7] 495 57 81 I 53 782 39 145 36 829 85 342 40406 35883 78 252 70 088 70 621 67 153 05 737 40933 91 075 91 259 82 100 13042 74 152 65927 16118 96032 64 150 23982 97868 52401 05 115 63754 71982 11067 83593 63678 05210 71700 76771 84474 02225 87644 33026 71 799 02423 88087 54114 53990 66397 80579 42 517 78 181 39251 09467 585 586 587 5GB 589 590 591 592 593 594 595 81167 702R4 21539 32397 39848 35083 66477 43836 41014 97025 93225 08511 63096 26628 73012 12543 76269 99708 02629 49845 73677 19193 14924- 57236 95564 15010 59667 73 773 78515 02624 99 744· 135R5 33 746 58 771 94785 62628 99585 11363 80832 59979 09444- 78700 02 596 85984 69138 16913 964-75 93283 ]8625 77086 45911 39746 64722 39938 43930 54619 00302 5038475249 53 163 27236 57 532 89514 26883 75714 16479 08415 03415 95439 80714 52 555 47266 48071 28 000 45011 26733 52651 89059 64 844 80910 299Bl 61 200 96036 62600 69165 97237 22 361 55276 96 190 67 132 01676 20068 39902 78 750 83 362 91 752 56530 95927 94973 84 162 57815 38487 82 190 596 597 598 599 600 02 738 83669 43 028 26264 08432 33 BUREAU OF INDIAN STANDARDS Headquarters: Manak Bhavan, 9 Bahadur Shah Zafar Marg, NEW DELHI 110002 Telephones: 323 0131, 323 3375, 323 9402 Fax :» 91 011 3234062, 3239399, 3239382 E-Mail: bis@vsnl.com Website: http://www.bis.org.in Central Laboratory: Plot No. 20/9, Site IV, Sahibabad Industrial Area, Sahibabad 201010 Telephone 4770032 Regional Offices: Central: Manak Bhavan, 9 Bahadur Shah Zafar Marg, NEW DELH,I 110002 *Eastern : 1/14 CIT Scheme VII, V.I.P. Road, Kankurgachi, CALCUTIA 700054 Northern: SCQ 335-336, Sector 34·A, CHANDIGARH 160022 Southern: C.I.T. Campus, IV Cross Road, CHENNAI 600113 tWestern : Manakalaya, E9, MIDC, Behind Marol Telephone Exchange, Andheri (East), MUMBAI 400093 32376 17 3378662 6038 43 254 13 15 8329295 Branch Offices: 'Pushpak', Nurmohamed Shaikh Marg, Khanpur, AHMEDABAD 380001 Peenya Industrial Area, 1st Stage, Bangalore-Tumkur Road, BANGALORE 560058 Commercial-cum-Qffice Complex, Opp. Dushera Maidan, E-5 Arera Colony, Bittan Market, BHOPAL 462016 62-63, Ganga Nagar, Unit VI, BHUBANESHWAR 751001 5th Floor, Kovai Towers, 44 8ala Sundaram Road, COIMBATORE 641018 Plot Nd. 58, Neelam Bata Road, NIT, FARIDABAD 121001 Savitri Complex, 116 G.T. Road, GHAZIABAD 201001 53/5 Ward No. 29, R.G. Barua Road, 5th By-lane, Apurba Sinha Path, GUWAHATI781003 5-8-56C, L.N. Gupta Marg, Nampally Station Road, HYDERABAD 500001 E-52, Chitaranjan Marg, C-Scheme, JAIPUR 302001 117/418 8, Sarvodaya. Nagar, KANPUR 208005 Seth Shawan, 2nd Floor, Behind Leela Cinema, Naval Kist:lore Road, LUCKNOW 226001 NIT Building, Second Floor, Gokulpat Market, NAGPUR 440010 Mahabir Bhawan, 1st Floor, Ropar Road, NALAGARH 174101 Patliputra Industrial Estate, PATNA 800013 First Floor, Plot Nos. 657-660, Market Yard, Gultekdi, PUNE 411037 'Sahajanand House' 3rd Floor, Bhaktinagar Circle, 80 Feet Road. RAJKOT 360002 I.C. No. 14/1421, University P.O. Palayam, THIRUVANANTHAPURAM 695034 ·Sales Office is at 5 Chowringhee Approach, P.O.Princep Street, CALCUTTA 700072 550 1348 8394955 723452 403627 21 88 35 54282 61 471 1998 54 11 37 320 10 84 3738 79 21 68 76 21 8923 5251 71 2 1451 262808 4268659 37 82 51 32 21 04 237 10 85 3096528 t Sales Office is at Novelty Chambers, Grant Road, MUMBAI 400007 Printed at Simco Printing Press, Delhi